Neither of these visions is entirely true. On Easter Sunday my French bishop is more than happy to advise his flock on how they should vote and the Catholic church in France is doing well and is still very powerful in a discreet way. And while Americans are accustomed to their presidents and political leaders using biblical references and asking that "God bless America" in their speeches, not everyone is happy about that. How well this rhetoric translates into votes depends greatly on the region.
Both countries see religion through very different lenses and came to the separation of church and state by very different paths. In France it was about curbing the power of one establishment religion, Catholicism, and allowing for "la liberté de conscience." It was a 1905 law that got the French government out of the religion business and the church out of the government while guaranteeing freedom of conscience. It was a subject of enormous debate at the time and Pope Pius X was firmly against it. It is still being debated. This article from Le Monde argues that this law is being used today to limit the religious freedom of some (mostly immigrants and their descendants) by attempting to entirely eliminate the visibility of religion in the public sphere - something the author of the piece says it was never intended to do. He also points out (and I did not know this) that there is one region in France where the national law concerning the separation of church and state does not apply. Alsace-Moselle is exempt from both the 1905 and the Jules Ferry law. The region has four official approved sects, their pastors/priests/rabbis are paid by the local government, and the public schools include religion in their curriculum. This is an exception but an important one.
That is not the only issue. As France enters the 21st century, it is becoming more and more religiously diverse: not only more Moslems but also the missionary activities of other sects like the Jehovah's Witnesses. Some French may be uncomfortable with Islam but they are even more so with sects that are disturbing to them because they are so very far from the mainstream: Mormons or Pentacostals. But I would argue that these debates do not demonstrate that the French are against religion. On the contrary, it is alive and well here and the churches I've been to are very well attended. But religion is not part of the language of politics. Even the most conservative political parties like the Front National support the principle of "laïcité."
From the founding of the United States the situation was entirely different - it was about accommodating many sects and the founders didn't have much choice in the matter. In the Colonial era (pre-independence) alone there was already a lot of variety: Puritans, Quakers, Catholics, Anglicans, Deists and so on. Many of the colonies were, in fact, founded by religious people whose goals were not in any way shape or form to create secular societies. As Andrew Preston says in his superb book, Sword of the Spirit, Shield of Faith, about religion in America and its influence on foreign policy, "Crucially, colonial America was established as a Reformation society, founded by Protestant radicals who took refuge from the religious wars and economic crises of Europe." Today the religious realm is even more chaotic, a "deregulated marketplace of faith." This site lists the top 25 denominations in the U.S. and Canada (out of 217) and says that in addition to that, in 2010 there were at least 35,000 independent (non-denominational) churches serving more than 12 million people. And while the focus is all too often on the religious political and social conservatives, one could argue that there are just as many churches and parishioners that swing to the Left of the political spectrum.
Preston points out that whatever the opinion of the elites in the U.S. (from the founding of the nation to the present day) they cannot ignore the fact that so many Americans do believe and find it perfectly normal that their views, informed by their faith, are included in national debates. They in turn expect their political leaders to answer back using terms and references that respect those beliefs. It is a top-down, bottom-up dialogue that is very disconcerting to many outside the U.S. (and to some Americans as well). I think it is fair to say that the majority of Americans simply do not believe that separation of church and state means that faith has no place in the public sphere. Religion in the U.S. is a part of the language of politics.
Is this a good or a bad thing? I find that I don't much care for the question. Context is everything. Traditions and histories differ. What makes sense in one world makes no sense at all in another. I find there is some merit to the purging of religion from public discourse in worlds that have known incredibly strife because of it. If we can't agree (and are willing at times to kill each other over it) perhaps it's best that we not discuss it at all. On the other hand, where would the U.S. be today without the Reverend Martin Luther King Jr? Or for that matter does anyone really think that France would be a better place today if l'abbé Pierre had been told to take his person and his beliefs back to the monastery? I cannot speak for anyone but myself but I personally would not like to have been deprived of the pleasure of reading these words addressed to the American public by the president in 1865:
With malice toward none; with charity for all; with firmness in the right, as God gives us to see the right, let us strive on to finish the work we are in; to bind up the nation's wounds; to care for him who shall have borne the battle, and for his widow, and his orphan--to do all which may achieve and cherish a just and lasting peace, among ourselves, and with all nations.